Just wondering if I could have some help proving that there is an injection from the set of all integers to the set of all reals. I think I need to use the identity function but not too sure

Thanks!

Oct 9th 2009, 03:32 PM

tonio

Quote:

Originally Posted by garrett12

Just wondering if I could have some help proving that there is an injection from the set of all integers to the set of all reals. I think I need to use the identity function but not too sure

Thanks!

Is the identity function an injection? Well, there you go.

Tonio

Oct 10th 2009, 01:16 AM

garrett12

Thanks for the reply. I get how the identity function is an injection, but I don't quite get the formal way to write a proof saying that the injection exists between Z and R. Any hints on the formal definition?

Oct 10th 2009, 03:12 AM

Defunkt

Quote:

Originally Posted by garrett12

Thanks for the reply. I get how the identity function is an injection, but I don't quite get the formal way to write a proof saying that the injection exists between Z and R. Any hints on the formal definition?

is an injection if

Let , the identity function. Let such that . Then, obviously, and so is an injection.

Oct 10th 2009, 07:13 PM

garrett12

Thanks Defunkt, that was what I was looking for. Just another quick question, along the same lines. If I'm asked to prove that the injection relationship is reflexive and transitive - but i'm not given a domain or range with it - how would i go about proving this? Can I just make up my own domain and range that it works for and then prove that?

Thanks

Oct 10th 2009, 11:52 PM

Defunkt

Quote:

Originally Posted by garrett12

Thanks Defunkt, that was what I was looking for. Just another quick question, along the same lines. If I'm asked to prove that the injection relationship is reflexive and transitive - but i'm not given a domain or range with it - how would i go about proving this? Can I just make up my own domain and range that it works for and then prove that?

Thanks

You don't "make up" your domain and range -- you have to show that it holds for any domain and range, ie. that:

if there exist injections , then there exists an injection (this is transitivity).

and that for any set X, there exists an injection

Can you finish from here?

Oct 11th 2009, 01:18 AM

garrett12

Quote:

Originally Posted by Defunkt

You don't "make up" your domain and range -- you have to show that it holds for any domain and range, ie. that:

if there exist injections , then there exists an injection (this is transitivity).

and that for any set X, there exists an injection

Can you finish from here?

I have trouble coming up with the formal ways of explaining it..

For transitivity:

If f:X->Y and g:Y->Z are injections..
For all x in X, y in Y and z in Z s.t f(x) = y and f(y) = z, ???

I get that xRz as no other unique value from X can relate to z but can't explain it :(

For symmetric:

Should the identity function be used again here? Seems like it should be

Oct 11th 2009, 02:00 AM

Defunkt

Quote:

Originally Posted by garrett12

I have trouble coming up with the formal ways of explaining it..

For transitivity:

If f:X->Y and g:Y->Z are injections..
For all x in X, y in Y and z in Z s.t f(x) = y and f(y) = z, ???

I get that xRz as no other unique value from X can relate to z but can't explain it :(

For symmetric:

Should the identity function be used again here? Seems like it should be

To show transitivity, you need to show that given that there are injections , there is an injection .

Easily enough, let , then . Can you show that h is an injection?

For reflexivity: The identity function is an injection from X to X.

The relation is not symmetric, though. Can you see why?

Oct 11th 2009, 12:24 PM

garrett12

Quote:

Originally Posted by Defunkt

To show transitivity, you need to show that given that there are injections , there is an injection .

Easily enough, let , then . Can you show that h is an injection?

For reflexivity: The identity function is an injection from X to X.

The relation is not symmetric, though. Can you see why?

Could I use the counterexample of X={1,2,3} Y={7,8,9} and say that (1,3) is in the relation, but (3,1) is not?

And would a correct counterexample for antisymmetry be X={1,2,3} Y={1,2,3,4} and say that (1,2) is in the relation and (2,1) is in the relation but 2 != 1?

Oct 11th 2009, 12:52 PM

Defunkt

Quote:

Originally Posted by garrett12

Could I use the counterexample of X={1,2,3} Y={7,8,9} and say that (1,3) is in the relation, but (3,1) is not?

And would a correct counterexample for antisymmetry be X={1,2,3} Y={1,2,3,4} and say that (1,2) is in the relation and (2,1) is in the relation but 2 != 1?

No, that is incorrect. The relation is between whole sets, ie. if then is in the relation and so is since there is an injection from X to Y and also an injection from Y to X.

A proper counterexample to symmetry would be . Obviously, there is an injection from X to Y however there is no injection from Y to X, and so would be in the relation but would not be.

Can you come up with an example to anti-symmetry now?

Oct 12th 2009, 03:52 AM

garrett12

Quote:

Originally Posted by Defunkt

No, that is incorrect. The relation is between whole sets, ie. if then is in the relation and so is since there is an injection from X to Y and also an injection from Y to X.

A proper counterexample to symmetry would be . Obviously, there is an injection from X to Y however there is no injection from Y to X, and so would be in the relation but would not be.

Can you come up with an example to anti-symmetry now?

Right, well for anti-symmetry I think that you just need the two sets to have the same cardinality and different elements to prove that it is not antisymmetric.

E.g X = {1,2,3}, Y={4,5,6}

There exists an injection from X->Y and Y->X, however X != Y. Hence the relation is not antisymmetric.

?

Oct 12th 2009, 07:42 AM

Defunkt

Quote:

Originally Posted by garrett12

Right, well for anti-symmetry I think that you just need the two sets to have the same cardinality and different elements to prove that it is not antisymmetric.

E.g X = {1,2,3}, Y={4,5,6}

There exists an injection from X->Y and Y->X, however X != Y. Hence the relation is not antisymmetric.